Piecewise suboptimal control laws for differential games

This paper is concerned with optimal parameter selection in differential games. Necessary and sufficient conditions are derived for the existence of a saddle point for a general two-person zero-sum differential game when one or both the players use suboptimal control laws of specified form. The specified forms for the controls consist of weighted sums of the state variables, the weighting factors being products of known time-varying functions and of piecewise-constant functions to be determined in an optimal manner. The controls which are formed in this way are referred to as piecewise control laws. The time intervals associated with the piecewise control laws can be different for each player. The general results are applied to linear-quadratic games, and for this class of differential games, an additional development is given to obtain piecewise control law parameters that are independent of initial conditions, so that a saddle point with respect to the expected value of the performance index is obtained. Consideration has also been given to the problem of optimizing the gain change points. The results are applied to scalar and vector dynamic systems, and numerical solutions are presented.